In the state of the art, many applications recover time variations of parameters, such as distance and velocity, through the effect on the phase of test signals, pressure, temperature, distance, electric field strength, magnetic field strength, strain, and acoustic fields being some of those parameters. Many instruments that are designed for those purposes are based on detecting the phase of a return periodic "probe" signal, which may be acoustic, radio frequency or optical but is limited by the amount of data that is processed, to avoid complexity. The time varying parameter is "band limited" by the nature of the parameter variation (e.g. the temperature change is limited by heat capacity and heat source-sink strength), which, following accepted signal processing concepts, imposes a fundamental sampling rate requirement for "unambiguously" recovering the parameter's variation with time. The requirement, which is well understood, is most often referred to as the "Nyquist" rate. The parameter variation, which forms information that may be seen in terms of quadrature signals I(t) and Q(t) that are proportional to the sine and cosine of the "carrier" plus the phase shift .phi.(t), is best expressed by the following equations for the case in which the parameter is distance: EQU I(t)=I.sub.0 +I.sub.1 .multidot.COS (2.pi.f.sub.0 t+k(t).multidot.X(t)+.phi..sub.0) 1) EQU Q(t)=I.sub.0 +I.sub.1 SIN (2.pi.f.sub.0 t+k(t).multidot.X(t)+.phi..sub.0)2) EQU .phi.(t)=k(t).multidot.X(t)+.phi..sub.0 3)
in which k(t) is the propagation constant (radians/distance), X(t) is the position at time t, .phi..sub.0 is a constant value for phase, inferred from the measurements of the phase .phi.(t) and f.sub.0 is the frequency (Hz) of the carrier. In these equations, k depends on the material parameters, and variations in k with time may be measured indirectly by the variation in phase, assuming that distance does not change. The minimum rate of simultaneous sampling of the quadrature signals I(t) and Q(t) is set by the requirement that the recovered phase .phi.(t) must not become ambiguous (i.e. undersampled) very often or unrecoverable information loss will appear. Occasional ambiguities can be treated in post-processing by the heuristic requirement that the recovered signal must not have discrete jumps-but only if the signal to noise ratio is high. The minimum rate is expressible in terms of the expected peak rate of change of phase (frequency) deviation from the carrier frequency and the quadrature pair I(t) and Q(t) must be sampled in less than the time required for the phase to advance or retard by a half-cycle relative to the carrier. This is equivalent to the Nyquist rate requirement on the frequency. Following conventional wisdom, sampling would take place at twice that rate.
The minimum Nyquist sampling rate is proportional to the product of the maximum phase shift and the frequency of variation of the parameter. In the usual case of a band limited parameter variation, a power spectral estimate is used to calculate the effective or design value of that product. A large amplitude is desirable because noise limits the precision with which the parameter can be estimated through the phase shift. This occurs because noise introduces uncertainty in the recovery of the phase shift on the fundamental measurement interval (0-360 degrees). If the phase shift is allowed to extend beyond this interval, the phase shifted signal bandwidth increases while the noise remains fixed. Similarly, high bandwidth is desirable to increase utility to a larger set of parameters (increased dynamic range).
Engineering practice follows two methods: limit the measurement of data per sample (using reduced phase resolution on each sample), allowing large amplitude-frequency products but limited in dynamic range; or reduce the allowable bandwidth or the amplitude of the parameters to provide higher resolution sampling at acceptable sampling rates. The first method is typically used for machine tool motion control; the second is more often used for optical testing of components. Neither method provides especially uniform high dynamic range capability with system sampling at the Nyquist rate set by the parameter band limit. The first method must sample at many times the parameter variation bandwidth set by the Nyquist rate. The second method may approach the Nyquist rate, but only as the dynamic range gets poorer, when the phase amplitude is then generally limited to less than .+-.180 degrees.